Coalgebras provide effective models of data structures and state-transition systems. A virtual covariety is a class of coalgebras closed under coproducts, images of coalgebraic morphisms, and subcoalgebras defined by split equalisers. A covariety has the stronger property of closure under all subcoalgebras, and is behavioural if it is closed under domains of morphisms, or equivalently under ima...

Coalgebraic speciications are used to formally describe the behaviour of classes in object-oriented languages. In this paper, a general notion of reenement between two such coalgebraic speciications is deened, capturing the idea that one \concrete" class speciication realises the behaviour of the other, \abstract" class speciication. Two (complete) proof-techniques are given to establish such r...

Recent progress on defining abstract trace semantics for coalgebras rests upon two observations: (i) coalgebraic bisimulation for deterministic automata coincides with trace equivalence, and (ii) the classical powerset construction for automata determinization instantiates the generic idea of lifting a functor to the Eilenberg-Moore category of an appropriate monad T. We take this approach one ...

In this paper, we model fresh names in the π-calculus using abstractions w.r.t. a new binding operator θ. Both the theory and the metatheory of the π-calculus benefit from this simple extension. The operational semantics of this new calculus is finitely branching. Bisimulation can be given without mentioning any constraint on names, thus allowing for a straightforward definition of a coalgebrai...

Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with ‘iterative alg...

We relate three different, but equivalent, ways to characterise behavioural equivalence for set coalgebras. These are: using final coalgebras, using coalgebraic languages that have the HennessyMilner property and using coalgebraic languages that have “logical congruences”. On the technical side the main result of our paper is a straightforward construction of the final T -coalgebra of a set fun...

The coalgebraic view on classes and objects is elaborated to include inheritance. Inheritance in coalgebraic speciication (of classes) will be understood dually to parametrization in algebraic speciication. That is, inheritance involves restriction (specialization), where parametrization involves extension. And cofree constructions are \best" restrictions, like free constructions are \best" ext...

A notion of reenement is deened in the context of coalgebraic spec-iication of classes in object-oriented languages. It tells us when objects in a \concrete" class behave exactly like (or: simulate) objects in an \abstract" class. The deenition of reenement involves certain selection functions between procedure-inputs and attribute-outputs, which gives this notion considerable exibility. The co...

Iterative theories introduced by Calvin Elgot formalize potentially infinite computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with “iterative algebras”, i. e., algeb...

This paper provides a coalgebraic modelling of Head-Driven Phrase Structure Grammar. HPSG is a licensing theory in the sense that grammaticality is defined in terms of abstract grammar principles, well-formedness conditions on linguistic analyses rather then by a set of rules that would explicitely generate an analysis. Since coalgebras are very well suited for modelling licensing theories, we ...